Sierpinski gaskets in the final matrix computation

The following program transforms regions of the final matrix into images that resemble the Sierpinski triangle.

The final matrix can only be positive on the Sierpinski triangle and the final matrix is negative elsewhere. In essence, the only interesting regions of the final matrix are the points near the boundary of the Sierpinski triangle. Since the Sierpinski triangle has dimension $\log_{2}(3)\approx 1.585$, the Sierpinski triangle hardly fills up space. The following tables assign to each region a ternary coordinate such that only the regions on the Sierpinski triangle get a ternary coordinate. The regions outside the Sierpinski triangle are labelled E.

0

1

E

2

00

01

10

11

E

02

E

12

E

E

20

21

E

E

E

22

000

001

010

011

100

101

110

111

E

002

E

012

E

102

E

112

E

E

020

021

E

E

120

121

E

E

E

022

E

E

E

122

E

E

E

E

200

201

210

211

E

E

E

E

E

202

E

212

E

E

E

E

E

E

220

221

E

E

E

E

E

E

E

222

Warning: Large images will crash your browser. It is recommended that you set $m\leq 10$ at first in order to prevent browser crashes.

Notice: For some values, the image produced by this program may be empty simply because large regions of the Sierpinski triangle do not have any positive final matrix values and hence nothing interesting is happening in those regions.

$n:$ select a natural number $n\leq 48$

$m$: select $m\leq n$

Input a number $x$ in base $3$ such that $0\leq x < 3^{n-m}$ for the location.

Check here to overlay the pixel for $\textrm{FM}_{n}^{+}(i,j)$ with the pixel for $\textrm{FM}_{n}^{+}(i,j+2^{n-1})$. If this box is checked then make sure that $m < n$.